Existence of compact support solutions for a quasilinear and singular problem

Jacques Giacomoni (1) , Habib Mâagli (2) , Paul Sauvy (1)

1 LMAP - Laboratoire de Mathématiques appliquées de Pau
2 FST - Faculté des Sciences Mathématiques, Physiques et Naturelles de Tunis

Let ­$ \Omega$ be a $\C^2$ bounded domain of $R^N$, $N\geq2$. We consider the following quasilinear elliptic problem: $$ (P_{\lambda}) \begin{array} −\delta_p(u) = K(x)(¸u^q − u^r), in ­\Omega,\\ u = 0 on \partial\Omega, u\geq0 in ­\Omega, \end{array} $$ where p > 1 and $\Delta_p(u) = div(|\nabla u|^{p−2}\nabla u) denotes the p-Laplacian operator. In this paper, $\lambda>0$ is a real parameter, the exponents q and r satisfy −1 < r < q < p − 1 and $K : ­\Omega\rightarrow R$ is a positive function having a singular behaviour near the boundary $\partial\Omega$. Precisely, $K(x) =\delt(x)^{−k}L(\delta(x))$ in ­$\Omega$, with 0 < k < p, L a positive perturbation function and $\delta(x)$ the distance of x 2 ­ to $\partial\Omega$. By using a sub- and super-solution technique, we discuss the existence of positive solutions or compact support solutions of $(P_{\lambda})$ in respect to the blow-up rate k. Precisely, we prove that if k < 1 + r, $(P_{\lambda})$ has at least one positive solution for $\lambda> 0$ large enough, whereas it has only compact support solutions if $k\geq 1 + r$.

Mots clés: Compact support principle; P-Laplace problem; Sub and super solutions

Domaines: Equations aux dérivées partielles [math.AP]

Jacques Giacomoni, Habib Mâagli, Paul Sauvy. Existence of compact support solutions for a quasilinear and singular problem. 2011. ⟨hal-00581771⟩